A367530 - OEIS

%I #28 Jul 08 2024 10:38:10

%S 1,4,32,2081,671104,954448620,5744387279872,144115188176529540,

%T 14925010118699132241920,6338253001141163895983922592,

%U 10985355337065420437221545952731136,77433143050453552574825990883161180320096,2213872302702432822841084717014014514981767643136

%N The number of ways of tiling the n X n torus up to matrix transposition by a tile that is asymmetric with respect to matrix transposition.

%C The n X n torus is an n X n grid where two grids are considered the same if one can reach the other by cyclic shifting of rows and columns.

%H Peter Kagey, <a href="/A367530/a367530_2.pdf">Illustration of a(3) = 32</a>

%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-25.

%t A367530[n_] := 1/(2n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2 d))]]])

%Y Cf. A103488, A255015.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 13 2023